A sharp bound on the size of a connected matroid

This paper proves that a connected matroid M in which a largest circuit and a largest cocircuit have c and c* elements, respectively, has at most 1/2c c* elements. It is also shown that if e is an element of M and c(e) and c*( e) are the sizes of a largest circuit containing e and a largest cocircuit containing e, then \E(M)\ less than or equal to (c(e)-1)(c*(e) -1)+1. Both these bounds are sharp and the first is proved using the second. The second inequality is an interesting companion to Lehman's width-length inequality which asserts that the former inequality can be reversed for regular matro ids when c(e) and c(*e) are replaced by the sizes of a smallest circuit con taining e and a smallest cocircuit containing e. Moreover, it follows from the second inequality that if u and v are distinct vertices in a 2-connecte d loopless graph G, then \E(G)\ cannot exceed the product of the length of a longest (u, v)-path and the size of a largest minimal edge-cut separating u from v.